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Metric Geometries in an Axiomatic Perspective

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From Riemann to Differential Geometry and Relativity

Abstract

In his 1854 Habilitationsvortrag Riemann presented a new concept of space endowed with a metric of great generality, which, through specification of the metric, gave rise to the spaces of constant curvature. In a different vein, yet with a similar aim, J. Hjelmslev, A. Schmidt, and F. Bachmann, introduced axiomatically a very general notion of plane geometry, which provides the foundation for the elementary versions of the geometries of spaces of constant curvature. We present a survey of these absolute geometric structures and their first-order axiomatizations, as well as of higher-dimensional variants thereof. In the 2-dimensional case, these structures were called metric planes by F. Bachmann, and they can be seen as the common substratum for the classical plane geometries: Euclidean, hyperbolic, and elliptic. They are endowed with a very general notion of orthogonality or reflection that can be specialized into that of the classical geometries by means of additional axioms. By looking at all the possible ways in which orthogonality can be introduced in terms of polarities, defined on (the intervals of a chain of subspaces of) projective spaces, one obtains a further generalization: the Cayley-Klein geometries. We present a survey of projective spaces endowed with an orthogonality and the associated Cayley-Klein geometries.

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Notes

  1. 1.

    “As is well known, geometry assumes as given both the notion of space and the fundamental notions for constructions in space. If offers merely nominal definitions for these notions, whereas the essential determinations appear in the form of axioms. In the process, the relation between these assumptions remains obscure; we neither realize whether and to what extent their association is necessary, nor a priori, whether it is possible.” (all translations are by V. Pambuccian).

  2. 2.

    The question of the validity of the hypotheses of geometry in the infinitely small is connected with the question of the intrinsic reasons for the metric relations of space. It is in this last question, which may still be regarded as belonging to the doctrine of space, that the remark made above finds its application, viz. that in the case of a discrete manifold, the principle of its metric relations are already contained in the very notion of this manifold, whereas in the case of a continuous manifold, this principle must come from somewhere else. Thus either the underlying reality of space must form a discrete manifold, or else we must seek the reason for its metric relations outside it, in binding forces acting upon it.

  3. 3.

    See [111, 2.2.10] for more on the influence of Herbart.

  4. 4.

    A homogeneous continuum, that would allow indefinite divisibility and would thus achieve the infinitely small, cannot be encountered anywhere in nature. The infinite divisibility of the continuum is an operation existing only in thought, only an idea, which is refuted by our observations of nature and by the experience drawn from physics and chemistry.

  5. 5.

    Die Untersuchungen von Riemann und Helmholtz über die Grundlagen der Geometrie.

  6. 6.

    To treat any question that might arise in a manner which also allowed us to check whether its answer is possible by a different route with certain restricted means.

  7. 7.

    Throughout this paper metric will always refer to a structure with an orthogonality relation or in which one such relation can be defined. It is in no way related to metrics defined as distances with real values.

  8. 8.

    “bahnt eine bequeme Straße durch ...[das] Urwaldgestrüpp der Lobatschefskijschen Rechnungen” [42, p. 277].

  9. 9.

    In the sense that there are no boundaries to a line, that one can travel along one without ever reaching anything remotely resembling an end, or, in Euclid’s own formulation, in Postulate 2 of Book I of the Elements, it is always possible “To produce a finite straight line continuously in a straight line.”.

  10. 10.

    When space-constructions are extended toward the unmeasurably large, one must distinguish between unboundedness and infinitude; the former belongs to the realm of extension, the latter to the that of measure.

  11. 11.

    “stellen sich vom logischen Standpunkte aus gleichberechtigt neben die euklidische Geometrie” [42, p. 164].

  12. 12.

    “da sie zum Teil nicht für Messungen in der Außenwelt verwendbar sind.” [42, p. 164].

  13. 13.

    Russell’s question is rhetorical in nature. He answers it on the next page, pointing out that the work of von Staudt, with its introduction of coordinates in a metric-free manner, removes all doubts regarding the independence of projective coordinates from distances.

  14. 14.

    The axiom system inside group theory can be found, with \(n=2\), in Sect. 3.

  15. 15.

    In [7, p. 339] one finds the only known algebraic characterization and in [7, Satz 1 on p. 286] a geometric characterization of these point-sets. Both are far from the specificity obtained in the actual solution of the Umkehrproblem for restricted classes of metric planes.

  16. 16.

    This kind of “concurrence” of three lines will be referred to as “the three lines lie in a pencil”.

  17. 17.

    A different axiomatization for the geometry obtained by adding \(\mathbf{E}_3\) has been provided in [74].

  18. 18.

    i.e., the elements \((\alpha \vee \varepsilon _k) \wedge \varepsilon _{k+1}\) and \((\beta \vee \varepsilon _k) \wedge \varepsilon _{k+1}\) (if \(\wedge \) and \(\vee \) denote the lattice operations).

  19. 19.

    This concept of a dilatation generalizes the notion of a dilatation which is given in incidence geometry (as a transformation which preserves direction) and in similarity geometry (as a transformation which preserves circles resp. the angular measure).

  20. 20.

    cp. Klein [42], Bachmann [7], Klopsch [44], Hessenberg and Diller [32] and Karzel and Kroll [38].

  21. 21.

    According to Sylvester’s law of inertia all maximal positive definite subspaces of a (real) quadratic space V, i.e., of a vector space endowed with a quadratic form, have the same dimension, which is called the signature of the quadratic space (the term “signature” is used in the literature in different ways; we follow Snapper and Troyer [99]). The signature of a subspace U of V is the signature of U with respect of the restriction of the quadratic form of V to U, see [108].

  22. 22.

    or more precisely a non-doubly hyperbolic Cayley-Klein group.

  23. 23.

    We recall from Sect. 3: Elements \(a, b, c, \ldots \) of S are called lines and elements \(A, B, C, \ldots \) of P points. The “stroke relation” \(\alpha \! \mid \! \beta \) is an abbreviation for the statement that \(\alpha , \beta \) and \(\alpha \beta \) are involutory elements (i.e., group elements of order 2). The statement \(\alpha , \beta \mid \delta \) is an abbreviation of \(\alpha \! \mid \! \delta \) and \(\beta \! \mid \! \delta \). A point A and a line b are incident if \(A \! \mid \! b\). Lines \(a, b \in S\) are orthogonal if \(a \! \mid \! b \). A quadrangle is a set of four points A, B, C, D and four lines a, b, c, d with \(a \mid A, B\) and \(b \mid B, C\) and \(c \mid C, D\) and \(d \mid D, A\).

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Authors and Affiliations

  1. School of Mathematical and Natural Sciences, Arizona State University — West Campus, P. O. Box 37100, Phoenix, AZ, 85069-7100, USA

    Victor Pambuccian

  2. Institut für Mathematikdidaktik, Universität zu Köln, Gronewaldstraße 2, 50931, Cologne, Germany

    Horst Struve

  3. SIGNAL IDUNA Gruppe, Joseph-Scherer-Straße 3, 44139, Dortmund, Germany

    Rolf Struve

Authors
  1. Victor Pambuccian
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  2. Horst Struve
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  3. Rolf Struve
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Correspondence to Victor Pambuccian .

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Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA

    Lizhen Ji

  2. University of Strasbourg, CNRS, Institut de Recherche Mathématique Avancée, Strasbourg Cedex, France

    Athanase Papadopoulos

  3. Department of Mathematics, Gakushuin University, Tokyo, Japan

    Sumio Yamada

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Pambuccian, V., Struve, H., Struve, R. (2017). Metric Geometries in an Axiomatic Perspective. In: Ji, L., Papadopoulos, A., Yamada, S. (eds) From Riemann to Differential Geometry and Relativity. Springer, Cham. https://doi.org/10.1007/978-3-319-60039-0_14

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